Add Recursive Staircase Problem.

Author: trekhleb

README.md

@@ -139,6 +139,7 @@ a set of rules that precisely define a sequence of operations.
   * `B` [Jump Game](src/algorithms/uncategorized/jump-game) - backtracking, dynamic programming (top-down + bottom-up) and greedy examples 
   * `B` [Unique Paths](src/algorithms/uncategorized/unique-paths) - backtracking, dynamic programming and Pascal's Triangle based examples 
   * `B` [Rain Terraces](src/algorithms/uncategorized/rain-terraces) - trapping rain water problem (dynamic programming and brute force versions)
+  * `B` [Recursive Staircase](src/algorithms/uncategorized/recursive-staircase) - count the number of ways to reach to the top (4 solutions)
   * `A` [N-Queens Problem](src/algorithms/uncategorized/n-queens)
   * `A` [Knight's Tour](src/algorithms/uncategorized/knight-tour)
 
@@ -151,6 +152,7 @@ algorithm is an abstraction higher than a computer program.
 * **Brute Force** - look at all the possibilities and selects the best solution
   * `B` [Linear Search](src/algorithms/search/linear-search)
   * `B` [Rain Terraces](src/algorithms/uncategorized/rain-terraces) - trapping rain water problem
+  * `B` [Recursive Staircase](src/algorithms/uncategorized/recursive-staircase) - count the number of ways to reach to the top
   * `A` [Maximum Subarray](src/algorithms/sets/maximum-subarray)
   * `A` [Travelling Salesman Problem](src/algorithms/graph/travelling-salesman) - shortest possible route that visits each city and returns to the origin city
   * `A` [Discrete Fourier Transform](src/algorithms/math/fourier-transform) - decompose a function of time (a signal) into the frequencies that make it up
@@ -178,6 +180,7 @@ algorithm is an abstraction higher than a computer program.
   * `B` [Jump Game](src/algorithms/uncategorized/jump-game)
   * `B` [Unique Paths](src/algorithms/uncategorized/unique-paths)
   * `B` [Rain Terraces](src/algorithms/uncategorized/rain-terraces) - trapping rain water problem
+  * `B` [Recursive Staircase](src/algorithms/uncategorized/recursive-staircase) - count the number of ways to reach to the top
   * `A` [Levenshtein Distance](src/algorithms/string/levenshtein-distance) - minimum edit distance between two sequences
   * `A` [Longest Common Subsequence](src/algorithms/sets/longest-common-subsequence) (LCS)
   * `A` [Longest Common Substring](src/algorithms/string/longest-common-substring)

src/algorithms/uncategorized/recursive-staircase/README.md

@@ -0,0 +1,21 @@
+# Recursive Staircase Problem
+
+## The Problem
+
+There are `n` stairs, a person standing at the bottom wants to reach the top. The person can climb either `1` or `2` stairs at a time. _Count the number of ways, the person can reach the top._
+
+![](https://cdncontribute.geeksforgeeks.org/wp-content/uploads/nth-stair.png)
+
+## The Solution
+
+This is an interesting problem because there are several ways of how it may be solved that illustrate different programming paradigms.
+
+- [Brute Force Recursive Solution](./recursiveStaircaseBF.js) - Time: `O(2^n)`; Space: `O(1)`
+- [Recursive Solution With Memoization](./recursiveStaircaseMEM.js) - Time: `O(n)`; Space: `O(n)`
+- [Dynamic Programming Solution](./recursiveStaircaseDP.js) - Time: `O(n)`; Space: `O(n)`
+- [Iterative Solution](./recursiveStaircaseIT.js) - Time: `O(n)`; Space: `O(1)` 
+
+## References
+
+- [On YouTube by Gayle Laakmann McDowell](https://www.youtube.com/watch?v=eREiwuvzaUM&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8&index=81&t=0s)
+- [GeeksForGeeks](https://www.geeksforgeeks.org/count-ways-reach-nth-stair/)

src/algorithms/uncategorized/recursive-staircase/__test__/recursiveStaircaseBF.test.js

@@ -0,0 +1,18 @@
+import recursiveStaircaseBF from '../recursiveStaircaseBF';
+
+describe('recursiveStaircaseBF', () => {
+  it('should calculate number of variants using Brute Force solution', () => {
+    expect(recursiveStaircaseBF(-1)).toBe(0);
+    expect(recursiveStaircaseBF(0)).toBe(0);
+    expect(recursiveStaircaseBF(1)).toBe(1);
+    expect(recursiveStaircaseBF(2)).toBe(2);
+    expect(recursiveStaircaseBF(3)).toBe(3);
+    expect(recursiveStaircaseBF(4)).toBe(5);
+    expect(recursiveStaircaseBF(5)).toBe(8);
+    expect(recursiveStaircaseBF(6)).toBe(13);
+    expect(recursiveStaircaseBF(7)).toBe(21);
+    expect(recursiveStaircaseBF(8)).toBe(34);
+    expect(recursiveStaircaseBF(9)).toBe(55);
+    expect(recursiveStaircaseBF(10)).toBe(89);
+  });
+});

src/algorithms/uncategorized/recursive-staircase/__test__/recursiveStaircaseDP.test.js

@@ -0,0 +1,18 @@
+import recursiveStaircaseDP from '../recursiveStaircaseDP';
+
+describe('recursiveStaircaseDP', () => {
+  it('should calculate number of variants using Dynamic Programming solution', () => {
+    expect(recursiveStaircaseDP(-1)).toBe(0);
+    expect(recursiveStaircaseDP(0)).toBe(0);
+    expect(recursiveStaircaseDP(1)).toBe(1);
+    expect(recursiveStaircaseDP(2)).toBe(2);
+    expect(recursiveStaircaseDP(3)).toBe(3);
+    expect(recursiveStaircaseDP(4)).toBe(5);
+    expect(recursiveStaircaseDP(5)).toBe(8);
+    expect(recursiveStaircaseDP(6)).toBe(13);
+    expect(recursiveStaircaseDP(7)).toBe(21);
+    expect(recursiveStaircaseDP(8)).toBe(34);
+    expect(recursiveStaircaseDP(9)).toBe(55);
+    expect(recursiveStaircaseDP(10)).toBe(89);
+  });
+});

src/algorithms/uncategorized/recursive-staircase/__test__/recursiveStaircaseIT.test.js

@@ -0,0 +1,18 @@
+import recursiveStaircaseIT from '../recursiveStaircaseIT';
+
+describe('recursiveStaircaseIT', () => {
+  it('should calculate number of variants using Iterative solution', () => {
+    expect(recursiveStaircaseIT(-1)).toBe(0);
+    expect(recursiveStaircaseIT(0)).toBe(0);
+    expect(recursiveStaircaseIT(1)).toBe(1);
+    expect(recursiveStaircaseIT(2)).toBe(2);
+    expect(recursiveStaircaseIT(3)).toBe(3);
+    expect(recursiveStaircaseIT(4)).toBe(5);
+    expect(recursiveStaircaseIT(5)).toBe(8);
+    expect(recursiveStaircaseIT(6)).toBe(13);
+    expect(recursiveStaircaseIT(7)).toBe(21);
+    expect(recursiveStaircaseIT(8)).toBe(34);
+    expect(recursiveStaircaseIT(9)).toBe(55);
+    expect(recursiveStaircaseIT(10)).toBe(89);
+  });
+});

src/algorithms/uncategorized/recursive-staircase/__test__/recursiveStaircaseMEM.test.js

@@ -0,0 +1,18 @@
+import recursiveStaircaseMEM from '../recursiveStaircaseMEM';
+
+describe('recursiveStaircaseMEM', () => {
+  it('should calculate number of variants using Brute Force with Memoization', () => {
+    expect(recursiveStaircaseMEM(-1)).toBe(0);
+    expect(recursiveStaircaseMEM(0)).toBe(0);
+    expect(recursiveStaircaseMEM(1)).toBe(1);
+    expect(recursiveStaircaseMEM(2)).toBe(2);
+    expect(recursiveStaircaseMEM(3)).toBe(3);
+    expect(recursiveStaircaseMEM(4)).toBe(5);
+    expect(recursiveStaircaseMEM(5)).toBe(8);
+    expect(recursiveStaircaseMEM(6)).toBe(13);
+    expect(recursiveStaircaseMEM(7)).toBe(21);
+    expect(recursiveStaircaseMEM(8)).toBe(34);
+    expect(recursiveStaircaseMEM(9)).toBe(55);
+    expect(recursiveStaircaseMEM(10)).toBe(89);
+  });
+});

src/algorithms/uncategorized/recursive-staircase/recursiveStaircaseBF.js

@@ -0,0 +1,27 @@
+/**
+ * Recursive Staircase Problem (Brute Force Solution).
+ *
+ * @param {number} stairsNum - Number of stairs to climb on.
+ * @return {number} - Number of ways to climb a staircase.
+ */
+export default function recursiveStaircaseBF(stairsNum) {
+  if (stairsNum <= 0) {
+    // There is no way to go down - you climb the stairs only upwards.
+    // Also if you're standing on the ground floor that you don't need to do any further steps.
+    return 0;
+  }
+
+  if (stairsNum === 1) {
+    // There is only one way to go to the first step.
+    return 1;
+  }
+
+  if (stairsNum === 2) {
+    // There are two ways to get to the second steps: (1 + 1) or (2).
+    return 2;
+  }
+
+  // Sum up how many steps we need to take after doing one step up with the number of
+  // steps we need to take after doing two steps up.
+  return recursiveStaircaseBF(stairsNum - 1) + recursiveStaircaseBF(stairsNum - 2);
+}

src/algorithms/uncategorized/recursive-staircase/recursiveStaircaseDP.js

@@ -0,0 +1,33 @@
+/**
+ * Recursive Staircase Problem (Dynamic Programming Solution).
+ *
+ * @param {number} stairsNum - Number of stairs to climb on.
+ * @return {number} - Number of ways to climb a staircase.
+ */
+export default function recursiveStaircaseDP(stairsNum) {
+  if (stairsNum < 0) {
+    // There is no way to go down - you climb the stairs only upwards.
+    return 0;
+  }
+
+  // Init the steps vector that will hold all possible ways to get to the corresponding step.
+  const steps = new Array(stairsNum + 1).fill(0);
+
+  // Init the number of ways to get to the 0th, 1st and 2nd steps.
+  steps[0] = 0;
+  steps[1] = 1;
+  steps[2] = 2;
+
+  if (stairsNum <= 2) {
+    // Return the number of ways to get to the 0th or 1st or 2nd steps.
+    return steps[stairsNum];
+  }
+
+  // Calculate every next step based on two previous ones.
+  for (let currentStep = 3; currentStep <= stairsNum; currentStep += 1) {
+    steps[currentStep] = steps[currentStep - 1] + steps[currentStep - 2];
+  }
+
+  // Return possible ways to get to the requested step.
+  return steps[stairsNum];
+}

src/algorithms/uncategorized/recursive-staircase/recursiveStaircaseIT.js

@@ -0,0 +1,31 @@
+/**
+ * Recursive Staircase Problem (Iterative Solution).
+ *
+ * @param {number} stairsNum - Number of stairs to climb on.
+ * @return {number} - Number of ways to climb a staircase.
+ */
+export default function recursiveStaircaseIT(stairsNum) {
+  if (stairsNum <= 0) {
+    // There is no way to go down - you climb the stairs only upwards.
+    // Also you don't need to do anything to stay on the 0th step.
+    return 0;
+  }
+
+  // Init the number of ways to get to the 0th, 1st and 2nd steps.
+  const steps = [1, 2];
+
+  if (stairsNum <= 2) {
+    // Return the number of possible ways of how to get to the 1st or 2nd steps.
+    return steps[stairsNum - 1];
+  }
+
+  // Calculate the number of ways to get to the n'th step based on previous ones.
+  // Comparing to Dynamic Programming solution we don't store info for all the steps but
+  // rather for two previous ones only.
+  for (let currentStep = 3; currentStep <= stairsNum; currentStep += 1) {
+    [steps[0], steps[1]] = [steps[1], steps[0] + steps[1]];
+  }
+
+  // Return possible ways to get to the requested step.
+  return steps[1];
+}

src/algorithms/uncategorized/recursive-staircase/recursiveStaircaseMEM.js

@@ -0,0 +1,44 @@
+/**
+ * Recursive Staircase Problem (Recursive Solution With Memoization).
+ *
+ * @param {number} totalStairs - Number of stairs to climb on.
+ * @return {number} - Number of ways to climb a staircase.
+ */
+export default function recursiveStaircaseMEM(totalStairs) {
+  // Memo table that will hold all recursively calculated results to avoid calculating them
+  // over and over again.
+  const memo = [];
+
+  // Recursive closure.
+  const getSteps = (stairsNum) => {
+    if (stairsNum <= 0) {
+      // There is no way to go down - you climb the stairs only upwards.
+      // Also if you're standing on the ground floor that you don't need to do any further steps.
+      return 0;
+    }
+
+    if (stairsNum === 1) {
+      // There is only one way to go to the first step.
+      return 1;
+    }
+
+    if (stairsNum === 2) {
+      // There are two ways to get to the second steps: (1 + 1) or (2).
+      return 2;
+    }
+
+    // Avoid recursion for the steps that we've calculated recently.
+    if (memo[stairsNum]) {
+      return memo[stairsNum];
+    }
+
+    // Sum up how many steps we need to take after doing one step up with the number of
+    // steps we need to take after doing two steps up.
+    memo[stairsNum] = getSteps(stairsNum - 1) + getSteps(stairsNum - 2);
+
+    return memo[stairsNum];
+  };
+
+  // Return possible ways to get to the requested step.
+  return getSteps(totalStairs);
+}